scholarly journals Explicit logarithmic formulas of special values of hypergeometric functions 3F2

2019 ◽  
Vol 22 (05) ◽  
pp. 1950040
Author(s):  
Masanori Asakura ◽  
Toshifumi Yabu
Keyword(s):  
Linear Combination ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Rational Numbers ◽  
Explicit Description ◽  
Generalized Hypergeometric Function ◽  
Algebraic Numbers ◽  
Nagoya Math ◽  
Special Values ◽  
Geometric Study

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of [Formula: see text] of the generalized hypergeometric function is a [Formula: see text]-linear combination of log of algebraic numbers if rational numbers [Formula: see text] satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it.

AN ALGEBRO-GEOMETRIC STUDY OF SPECIAL VALUES OF HYPERGEOMETRIC FUNCTIONS

2018 ◽  
Vol 236 ◽  
pp. 47-62
Author(s):  
MASANORI ASAKURA ◽  
NORIYUKI OTSUBO ◽  
TOMOHIDE TERASOMA
Keyword(s):  
Hypergeometric Functions ◽  
Sufficient Condition ◽  
Algebraic Numbers ◽  
Special Values ◽  
Special Value ◽  
Geometric Study

For a certain class of hypergeometric functions $_{3}F_{2}$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.

scholarly journals Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
T. A. Ishkhanyan ◽  
T. A. Shahverdyan ◽  
A. M. Ishkhanyan
Keyword(s):  
Power Series ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Power Series Expansion ◽  
Heun Equation ◽  
Generalized Hypergeometric Function ◽  
Gamma Functions ◽  
Heun Function ◽  
Finite Sum

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650064 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Contiguous Relations

The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

scholarly journals Integral representations of generalized Lauricella hypergeometric functions

1992 ◽  
Vol 15 (4) ◽  
pp. 653-657 ◽  
Author(s):  
Vu Kim Tuan ◽  
R. G. Buschman
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Appell Function

The generalized hypergeometric function was introduced by Srivastava and Daoust. In the present paper a new integral representation is derived. Similarly new integral representations of Lauricella and Appell function are obtained.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

Application of the E-operator to evaluate an infinite integral

1969 ◽  
Vol 65 (3) ◽  
pp. 725-730 ◽  
Author(s):  
F. Singh
Keyword(s):  
Finite Difference ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
General Nature ◽  
Difference Operators ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Infinite Integral

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

scholarly journals On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
N. Magesh ◽  
N. B. Gatti ◽  
S. Mayilvaganan
Keyword(s):  
Linear Operator ◽  
Linear Combination ◽  
Hypergeometric Function ◽  
Meromorphic Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Generalized Hypergeometric Function ◽  
Coefficient Estimates ◽  
Convex Linear Combination ◽  
Meromorphic Univalent Functions

We introduce and study a subclass ΣP(γ,k,λ,c) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ΣP(γ,k,λ,c) by fixing the second coefficient. Further, it is shown that the class ΣP(γ,k,λ) is closed under convex linear combination.

A new expansion formula for hypergeometric functions of two variables

1968 ◽  
Vol 64 (2) ◽  
pp. 413-416
Author(s):  
B. L. Sharma
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Expansion Formula ◽  
Generalized Hypergeometric Functions ◽  
Functions Of Two Variables ◽  
Function Of Two Variables

The main object of this paper is to derive an expansion formula for a generalized hypergeometric function of two variables in a series of products of generalized hypergeometric functions of two variables and a Meijer's G-function. The result established in this paper is the extension of the results recently given by Srivastava (5) and Verma (6). It is interesting to note that some interesting expansions can be derived from the result by specializing the parameters.

A NOTE ON THE ASYMPTOTIC EXPANSION OF GENERALIZED HYPERGEOMETRIC FUNCTIONS

2013 ◽  
Vol 12 (01) ◽  
pp. 107-115 ◽  
Author(s):  
HANS VOLKMER ◽  
JOHN J. WOOD
Keyword(s):  
Probability Theory ◽  
Asymptotic Expansion ◽  
Hypergeometric Function ◽  
Explicit Expression ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions

We use arguments from probability theory to derive an explicit expression for the asymptotic expansion of the generalized hypergeometric function for p = q and show that the coefficients in the expansion are polynomials in the parameters of the hypergeometric function. The idea behind this paper originates from the second author's Ph.D. thesis, where the case p = 2 is treated.

Generalized Neumann expansions involving hypergeometric functions

1967 ◽  
Vol 63 (2) ◽  
pp. 425-429 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Image Position

1. Making use of the familiar abbreviationlet us adopt a contracted notation for the generalized hypergeometric function AFB[x] and writewhere (a) denotes the sequence of parameters

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